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 The primary issue with life incarceration or the death penalty is that it’s carried out by the monopolistic gang called the state. There’s no right answer in our system. 
The third way is to abolish the state and let the free market justice system work ( ie Bob Murphy, Bruce Benson and David Friedman) The closer the state can approximate these systems by getting out of the way (like allowing discrimination against criminals) it would be better than what we have. 
 I agree with that

However, I think in practice it would more closely resemble the ancien regime, with punishments like death and exile much more common. I think the "outlaw" system of the American West worked very well, with criminals denied the protection of the law (and thus subject to death or other injury). I also think in a sytem of private law there would be armed self-defense, and thus another avenue by which the death of criminals would become much more common.

Murphy seems to think that a private law system would become nearly pacifist, but I strongly disagree. The ability to be soft requires very strong security which in turn requires that somewhere along the line there is iron determination. We are seeing exactly that tension unravel in modern Europe and the Anglophone countries. 
 It’s a tricky task to predict a free market! I think different cultures would look different. Followed for your thoughtful comment.